Problem: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{x^2 + 8x - 9}{x^2 - x}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 8x - 9}{x^2 - x} = \dfrac{(x + 9)(x - 1)}{(x)(x - 1)} $ Notice that the term $(x - 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x - 1)$ gives: $r = \dfrac{x + 9}{x}$ Since we divided by $(x - 1)$, $x \neq 1$. $r = \dfrac{x + 9}{x}; \space x \neq 1$